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Simplifying n * n + n = 75 Multiply n * n n2 + n = 75 Reorder the terms: n + n2 = 75 Solving n + n2 = 75 Solving for variable 'n'. Reorder the terms: -75 + n + n2 = 75 + -75 Combine like terms: 75 + -75 = 0 -75 + n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '75' to each side of the equation. -75 + n + 75 + n2 = 0 + 75 Reorder the terms: -75 + 75 + n + n2 = 0 + 75 Combine like terms: -75 + 75 = 0 0 + n + n2 = 0 + 75 n + n2 = 0 + 75 Combine like terms: 0 + 75 = 75 n + n2 = 75 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. n + 0.25 + n2 = 75 + 0.25 Reorder the terms: 0.25 + n + n2 = 75 + 0.25 Combine like terms: 75 + 0.25 = 75.25 0.25 + n + n2 = 75.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 75.25 Calculate the square root of the right side: 8.674675786 Break this problem into two subproblems by setting (n + 0.5) equal to 8.674675786 and -8.674675786.Subproblem 1
n + 0.5 = 8.674675786 Simplifying n + 0.5 = 8.674675786 Reorder the terms: 0.5 + n = 8.674675786 Solving 0.5 + n = 8.674675786 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 8.674675786 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 8.674675786 + -0.5 n = 8.674675786 + -0.5 Combine like terms: 8.674675786 + -0.5 = 8.174675786 n = 8.174675786 Simplifying n = 8.174675786Subproblem 2
n + 0.5 = -8.674675786 Simplifying n + 0.5 = -8.674675786 Reorder the terms: 0.5 + n = -8.674675786 Solving 0.5 + n = -8.674675786 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -8.674675786 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -8.674675786 + -0.5 n = -8.674675786 + -0.5 Combine like terms: -8.674675786 + -0.5 = -9.174675786 n = -9.174675786 Simplifying n = -9.174675786Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.174675786, -9.174675786}
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